The mathematics of wealth is kind of a tricky subject, and one that physicists tend to shy away from. They (physicists and mathematicians) tend to prefer physical processes that are as simple as possible, minimizing complexity. Of course chaos theory tackles complexity head on, but that's different: chaos can characterize extremely nonlinear systems and help us to understand why they are not predictive. But what we are really after in dynamical systems is to be able to predict! That's where the power lies.

For economics, the application of things like statistical mechanics is made difficult by the fact that when discussing wealth, we are not discussing conserved quantities. To illustrate the difficulty with concepts like wealth, say you start with $\$100$, and you loan me $\$20$. Now an amazing thing happens: you still have the $\$100$, because the debt owed is indeed an asset, and you do not have less wealth. But now, I have $\$20$, so the total wealth is now $\$120$. How would you construct a statistical mechanics model of something like a volume of gas when every time a molecule collides it makes more molecules with more energy? You would not be able to set up a partition function, there are no conserved quantities, and your model would be pretty limited. Welcome to economics. It's much harder than physics!

To understand the mathematics of wealth inequality, start with what is known as the "Yard Sale Model", based on a rather well defined economic system that describes the exchange of goods among people in a fixed population. In this model, there is no currency, but there is lots of trading among pairs of people. Imagine what would happen if every trade is fair (trading goods of equal value). Then, over time, everyone's net wealth will stay constant, and the net wealth of the community stays constant. Now imagine what would happen if the trades are sometimes not completely equal (by some measure). What happens over time is pretty interesting.

To be specific, say that we have 2 people, each with the same initial net wealth, which changes over time. Label them $A$ and $B$ such that $A$ is wealthier than $B$. Then for each transaction, you flip a coin, and if it comes up heads, $A$ gives an amount equal to $20\%$ of $B$'s wealth to $B$. If it comes up tails, then $A$ takes an amount equal to $17\%$ of $B$'s wealth, so that $B$ never loses more than their wealth. If we start with a population of some large number of people, then we can pair them off for each transaction, and then repeat the "play" (1 transaction for each pair) some number of times, each time assigning a random pairing. What happens when we do this trade many times?

In the simulation below, each person starts with $\$100$. You can choose the number of people trading, the default is 100, so the maximum amount of money that any one person can accumulate will be $\$100\times 100=\$10,000$, which happens when one person becomes the "oligarch". The text box labeled "gain" will allow you to set the percentage of the less wealthy person in the pair that they gain from the more wealthy if the coin flip goes their way, and loss will be the percentage of their wealth they loose to the more wealthy person if the coin flip goes the other way (they lose the flip).

The graph on the left displays the wealth of all people in the game, and the graph on the right displays the sequence of the wealth of a player chosen at random. The value in the "play" text box sets how many trade times between updating the plots. When you hit the "Run" button, the animation will run this game forever, updating as per the number set in the "plays" window, until you hit the "Stop" button. You will see how much time it takes for the oligarch to appear. Try hitting "Run" and then "Stop" a number of times until you get lucky enough to see the wealth distribution of one of the oligarchs.